1. Field of the Invention
The present invention relates to bandlimited synthesis of analog waveforms used for example in music synthesis.
2. Description of Related Art
Digital synthesis of any analog waveform must be bandlimited or the resulting signal will be aliased, as discussed in Brandt, “Hard sync without aliasing,” Proc. 2001 International Computer Music Conference, International Computer Music Association (ICMA), pp. 365-368; and in Stilson et al., “Alias-free digital synthesis of classic analog waveforms,” Proc. 1996 International Computer Music Conference, ICMA. Stilson et al. review several known methods for producing classic analog synthesizer waveforms and describe methods of generating sawtooth and pulse waveforms based on integrating bandlimited impulse trains (See Stilson et al., FIGS. 12 and 13). In practice, the integrators required for the Stilson et al. methods can cause problems such as DC offsets at startup. Therefore, it is desirable to find a method that avoids the integration step. Also, the Stilson et al. article does not discuss important techniques in music synthesis such as hard sync or other modulations.
Brandt describes a theoretical method of adding a minimum-phase bandlimited step, called a MinBLEP in the Brandt article, to introduce a bandlimited discontinuity into a waveform. Important details for practical implementation of the Brandt method are not described. For example, the minimum-phase bandlimited step can only be applied when a proper phase and amplitude of a discontinuity has been determined. The Brandt article fails to disclose the technique for determining such proper phase and amplitudes. Fundamentally, the minimum-phase bandlimited step has a DC bias and therefore creates a frequency-dependent DC error (See Brandt, FIG. 7).
In a hard sync technique, a slave waveform having a fundamental frequency is reset in response to a master waveform having a different fundamental frequency. This causes discontinuities in the slave waveform that have various amplitudes and occur at various times within respective sampling intervals. Such discontinuities cause aliasing in sampled data representations of analog waveforms. Japanese patent number P3399272, by Yamaha Corporation, entitled “Music Sound Issuing Device and Method of Issuing New Music Sound,” describes a method for synthesizing a sawtooth waveform based on a nonlinear waveshaping filter (See element 12 of FIG. 1 in P3399272). It further describes a hard sync technique that multiplies a slave waveform with a window function that is a function of the master waveform and the phase of the discontinuity (See FIG. 5 in P3399272). Nonlinear waveshaping methods and windowing methods like those described in P3399272 can reduce aliasing, but in practice still sound aliased.
It is desirable to provide improved procedures for producing bandlimited waveforms and bandlimited hard sync waveforms.